1.1 Functions and Their Representations

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1 Arkansas Tech University MATH 2914: Calculus I Dr. Marcel B. Finan 1.1 Functions and Their Representations Functions play a crucial role in mathematics. A function describes how one quantity depends on others. More precisely, when we say that a quantity y is a function of a quantity x we mean a rule that assigns to every possible value of x exactly one value of y. We call x the input and y the output. In function notation we write y = f(x) Since y depends on x, it makes sense to call x the independent variable and y the dependent variable. In applications of mathematics, functions are often representations of real world phenomena. Thus, the functions in this case are referred to as mathematical models. If the set of input values is a finite set then the models are known as discrete models. Otherwise, the models are known as continuous models. For example, if H represents the temperature after t hours for a specific day, then H is a discrete model. If A is the area of a circle of radius r then A is a continuous model. There are four common ways in which functions are presented and used: Verbally (by a description in words), numerically (by a table of values), Visually (by a graph), and algebraically (by a formula). Example The sales tax on an item is 6%. So if p denotes the price of the item and C the total cost of buying the item then if the item is sold at $ 1 then the cost is 1 + (0.06)(1) = $1.06 or C(1) = $1.06. If the item is sold at $2 then the cost of buying the item is 2 + (0.06)(2) = $2.12, or C(2) = $2.12, and so on. Thus we have a relationship between the quantities C and p such that each value of p determines exactly one value of C. In this case, we say that C is a function of p. Describes this function using words, a table, a graph, and a formula. Words: To find the total cost, multiply the price of the item by 0.06 and add the result to the price. Table: The chart below gives the total cost of buying an item at price p as a function of p for 1 p 6. 1

2 p C Graph: The graph of the function C is obtained by plotting the data in the above table. See Figure Formula: The formula that describes the relationship between C and p is given by C(p) = 1.06p. Figure So far, we have introduced rules between two quantities that define functions. Unfortunately, it is possible for two quantities to be related and yet for neither quantity to be a function of the other. Example Let x and y be two quantities related by the equation (a) Is x a function of y? Explain. (b) Is y a function of x? Explain. x 2 + y 2 = 4. (a) For y = 0 we have two values of x, namely, x = 2 and x = 2. So x is not a function of y. (b) For x = 0 we have two values of y, namely, y = 2 and y = 2. So y is not a function of x. 2

3 Next, suppose that the graph of a relationship between two quantities x and y is given. To say that y is a function of x means that for each value of x there is exactly one value of y. Graphically, this means that each vertical line must intersect the graph at most once. Hence, to determine if a graph represents a function one uses the following test: Vertical Line Test: A graph is a function if and only if every vertical line crosses the graph at most once. According to the vertical line test and the definition of a function, if a vertical line cuts the graph more than once, the graph could not be the graph of a function since we have multiple y values for the same x-value and this violates the definition of a function. Example Which of the graphs (a), (b), (c) in Figure represent y as a function of x? Figure By the vertical line test, (b) represents a function whereas (a) and (c) fail to represent functions since one can find a vertical line that intersects the graph more than once 3

4 Domain and Range of a Function If we try to find the possible input values that can be used in the function y = x 2 we see that we must restrict x to the interval [2, ), that is x 2. Similarly, the function y = 1 takes only certain values for the output, namely, y > 0. Thus, a function is often defined for certain values of x x 2 and the dependent variable often takes certain values. The above discussion leads to the following definitions: By the domain of a function we mean all possible input values that yield output values. Graphically, the domain is part of the horizontal axis. The range of a function is the collection of all possible output values. The range is part of the vertical axis. When finding the domain and range, there are simple basic rules to consider: The domain of all polynomial functions, i.e. functions of the form f(x) = a n x n + a n 1 x n a 1 x + a 0, where n is a non-negative integer, is the set of real numbers IR. Square root functions can not contain a negative underneath the radical symbol. Set the expression under the radical symbol greater than or equal to zero and solve for the variable. This will be your domain. Fractional functions, i.e. ratios of two functions, determine for which input values the numerator and denominator are not defined and the domain is everything else. For example, make sure not to divide by zero! Example Find, algebraically, the domain and the range of each of the following functions. Write your answers in interval notation: (a) y = πx 2 (b) y = 1 x 4 (c) y = x. (a) Since the function is a polynomial, its domain is the interval (, ). To find the range, solve the given equation for x in terms of y obtaining x = ± y π. Thus, x exists for y 0. So the range is the interval [0, ). (b) The domain of y = 1 x 4 consists of all numbers x such that x 4 > 0 or x > 4. That is, the interval (4, ). To find the range, we solve for x in terms of y > 0 obtaining x = x exists for all y > 0. Thus, the range y 2 is the interval (0, ). (c) The domain of y = x is the interval (, 0) (0, ). To find 4

5 the range, write x in terms of y to obtain x = 1 y 2. The values of y for which this later formula is defined is the range of the given function, that is, (, 2) (2, ) Piecewise Defined Functions Algebraically, it is possible to defined a function by a single formula or by multiple formulas. A piecewise defined function is a function defined by different formulas for different intervals of the independent variable. Example (The Absolute Value Function) (a) Show that the function f(x) = x is a piecewise defined function. (b) Graph f(x). (a) The absolute value function x is a piecewise defined function since { x for x 0 x = x for x < 0. (b) The graph is given in Figure Figure Example Sketch the graph of the piecewise defined function given by x + 4 for x 2 f(x) = 2 for 2 < x < 2 4 x for x 2. The following table gives values of f(x). 5

6 x f(x) The graph of the function is given in Figure Figure Symmetry For a given function f(x), the graph of the new function f(x) is the reflection of the graph of f(x) about the x axis. Example Graph the functions f(x) = 2 x and f(x) = 2 x on the same axes. The graph of both f(x) = 2 x and f(x) are shown in Figure Figure We know that the points x and x are on opposite sides of the x axis. So the graph of the new function f( x) is the reflection of the graph of f(x) about the y axis. If the reflection of the graph of f(x) about the y axis is the same as the 6

7 graph of f(x),i.e, f( x) = f(x), then we say that the graph of f(x) is symmetric about the y axis. We call such a function an even function. Example (a) Using a graphing calculator show that the function f(x) = (x x 3 ) 2 is even. (b) Now show that f(x) is even algebraically. (a) The graph of f(x) is symmetric about the y-axis so that f(x) is even. See Figure Figure (b) Since f( x) = ( x ( x) 3 ) 2 = ( x+x 3 ) 2 = [ (x x 3 )] 2 = (x x 3 ) 2 = f(x), f(x) is even Now, if the images f(x) and f( x) are of opposite signs i.e, f( x) = f(x), then the graph of f(x) is symmetric about the origin. In this case, we say that f(x) is odd. Alternatively, since f(x) = f( x), if the graph of a function is reflected first across the y axis and then across the x axis and you get the graph of f(x) again then the function is odd. Example (a) Using a graphing calculator show that the function f(x) = 1+x2 x x 3 (b) Now show that f(x) is odd algebraically. is odd. (a) The graph of f(x) is symmetric about the origin so that f(x) is odd. See Figure

8 (b) Since f( x) = Figure ( x)2 = 1+x2 = 1+x2 = f(x), f(x) is odd ( x) ( x) 3 x+x 3 (x x 3 ) A function can be either even, odd, both, or neither. Example (a) Show that the function f(x) = x 2 is even but not odd. (b) Show that the function f(x) = x 3 is odd but not even. (c) Show that the function f(x) = x + x 2 is neither odd nor even. (d) Is there a function that is both even and odd? Explain. (a) Since f( x) = f(x) and f( x) f(x), f(x) is even but not odd. (b) Since f( x) = f(x) and f( x) f(x), f(x) is odd but not even. (c) Since f( x) = x + x 2 ±f(x), f(x) is neither even nor odd. (d) We are looking for a function such that f( x) = f(x) and f( x) = f(x). This implies that f(x) = f(x) or 2f(x) = 0. Dividing by 2 to obtain f(x) = 0. This function is both even and odd and is the only one that is both Increasing and Decreasing Functions We say that a function is increasing if its graph climbs as x moves from left to right. That is, the function values increase as x increases. Symbolically, if a < b then we must have f(a) < f(b). A function f is said to be decreasing if its graph falls as x moves from left to right. This means that the function values decrease as x increases. Symbolically, if a < b then f(a) > f(b). Example Determine the intervals where the function, given in Figure 1.1.8, is increas- 8

9 ing and decreasing. Figure The function is increasing on (, 1) (1, ) and decreasing on the interval ( 1, 1) 9